Pluriadjoint bundles of polarized surfaces
Let L be an ample line bundle on a complex projective smooth surface X. The spannedness and the very ampleness properties of the pluriadjoint bundles \((K_ X\otimes L)^{\otimes t}\), \(t\geq 2\), are investigated by using Reider's theorem [\textit{I. Reider}, Ann. Math., II. Ser. 127, No.2, 309-316 (1988; Zbl 0663.14010)]. If (X,L) is not in a special list and \((X',L')\) denotes the reduction of (X,L), then (1) \((K_{X'}\otimes L')^{\otimes 3}\) is very ample; (2) \((K_{X'}\otimes L')^{\otimes 2}\) is ample and spanned and is very ample unless X contains an irreducible curve satisfying some special conditions; (3) the map associated with \(\Gamma ((K_{X'}\otimes L')^{\otimes 2})\) has degree \(\leq 4\), equality implying that \(\chi\) (\({\mathcal O}_ X)=0\) and \(g(L)=2\).
scientific article; zbMATH DE number 4175122